An infinite family of solvable and integrable quantum systems on a plane

An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.
Comment: 30 pages, Introduction extended, description of known integrals given, some statements clarified, one reference added, will be published in J Phys A (FTC)